非线性奇异二阶三点边值问题的对称正解(英文)

In this paper, the second-order three-point boundary value problem u(t) + λa(t)f(t, u(t)) = 0, 0 t 1,u(t) = u(1- t), u(0)- u(1) = u(12)is studied, where λ is a positive parameter, under various assumption on a and f, we establish intervals of the parameter λ, which yield the existence of positive solution, our proof based on Krasnosel'skii fixed-point theorem in cone.{u"(t)+λa(t)f(t,u(t))=0,0t1,u(t)=u(1-t),u′(0)-u′(1)=u(1/2)is studied,where A is a positive parameter,under various assumption on a and f,we establish intervals of the parameter A,which yield the existence of positive solution,our proof based on Krasnosel'skii fixed-point theorem in cone.     

The existence of nontrivial solutions to a semilinear elliptic system on without the ambrosetti-rabinowitz condition

In this paper, we prove the existence of at least one positive solution pair （u, v）∈ H1（RN） × H1（RN） to the following semilinear elliptic system {-△u＋u=f（x,v）,x∈RN,-△u＋u=g（x,v）,x∈RN （0.1）,by using a linking theorem and the concentration-compactness principle. The main conditions we imposed on the nonnegative functions f, g ∈C0（RN× R1） are that, f（x, t） and g（x, t） are superlinear at t = 0 as well as at t =＋∞, that f and g are subcritical in t and satisfy a kind of monotonic conditions. We mention that we do not assume that f or g satisfies the Ambrosetti-Rabinowitz condition as usual. Our main result can be viewed as an extension to a recent result of Miyagaki and Souto [J. Diff. Equ. 245（2008）, 3628-3638] concerning the existence of a positive solution to the semilinear elliptic boundary value problem {-△u＋u=f（x,u）,x∈Ω,u∈H0^1（Ω） where Ω ∩→RN is bounded and a result of Li and Yang [G. Li and J. Yang： Communications in P.D.E. Vol. 29（2004） Nos.5＆ 6.pp.925-954, 2004] concerning （0.1） when f and g are asymptotically linear.     

SIGN-CHANGING SOLUTIONS FOR SCHRODINGER EQUATIONS WITH VANISHING AND SIGN-CHANGING POTENTIALS

In this article, we study the existence of sign-changing solutions for the following SchrSdinger equation
-△u ＋ λV（x）u = K（x）｜u｜^p-2u x∈R^N, u→0 as ｜x｜→ ＋∞,
2N where N ≥ 3, λ〉 0 is a parameter, 2 〈 p 〈 2N/N-2, and the potentials V（x） and K（x） satisfy some suitable conditions. By using the method based on invariant sets of the descending flow, we obtain the existence of a positive ground state solution and a ground state sign-changing solution of the above equation for small λ, which is a complement of the results obtained by Wang and Zhou in [J. Math. Phys. 52, 113704, 2011].     

Symmetric Positive Solutions for a Singular Second-Order Three-Point Boundary Value Problem

In this paper, we consider the following second order three-point boundary value problem u″（t）＋a（t）f（u（t））=0,0〈t〈1,u（0）-u（1）=0,u＇（0）-u＇（1）=u（1/2）,where a ： （0, 1） → [0, ∞） is symmetric on （0, 1） and may be singular at t = 0 and t = 1, f ： [0, ∞） → [O, ∞） is continuous. By using Krasnoselskii＇s fixed point theorem ia a cone, we get some existence results of positive solutions for the problem. The associated Green＇s function for the three-point boundary value problem is also given.     

弱半正三阶三点边值问题的正解

The positive solutions are studied for the nonlinear third-order three-point boundary value problem u′″（t）=f（t,u（t））,a.e,t∈[0,1],u（0）=u′（η）=u″（1）=0, where the nonlinear term f（t, u） is a Caratheodory function and there exists a nonnegative function h ∈ L^1[0, 1] such that f（t, u） 〉 ≥-h（t）. The existence of n positive solutions is proved by considering the integrations of ＂height functions＂ and applying the Krasnosel＇skii fixed point theorem on cone.     

Existence and Global Asymptotic Behavior of Positive Solutions for Sublinear and Superlinear Fractional Boundary Value Problems

In this paper, the authors aim at proving two existence results of fractional differential boundary value problems of the form(P_(a,b)){D~αu(x) + f(x, u(x)) = 0, x ∈(0, 1),u(0) = u(1) = 0, D~(α-3)u(0) = a, u(1) =-b,where 3 α≤ 4, Dαis the standard Riemann-Liouville fractional derivative and a, b are nonnegative constants. First the authors suppose that f(x, t) =-p(x)t~σ, with σ∈(-1, 1)and p being a nonnegative continuous function that may be singular at x = 0 or x = 1and satisfies some conditions related to the Karamata regular variation theory. Combining sharp estimates on some potential functions and the Sch¨auder fixed point theorem, the authors prove the existence of a unique positive continuous solution to problem(P_(0,0)).Global estimates on such a solution are also obtained. To state the second existence result, the authors assume that a, b are nonnegative constants such that a + b 0 and f(x, t) = tφ(x, t), with φ(x, t) being a nonnegative continuous function in(0, 1)×[0, ∞) that is required to satisfy some suitable integrability condition. Using estimates on the Green's function and a perturbation argument, the authors prove the existence and uniqueness of a positive continuous solution u to problem(P_(a,b)), which behaves like the unique solution of the homogeneous problem corresponding to(P_(a,b)). Some examples are given to illustrate the existence results.     

Analysis of a Free Boundary Problem Modeling Tumor Growth

In this paper, we study a free boundary problem arising from the modeling of tumor growth. The problem comprises two unknown functions： R = R（t）, the radius of the tumor, and u = u（r, t）, the concentration of nutrient in the tumor. The function u satisfies a nonlinear reaction diffusion equation in the region 0 〈 r 〈 R（t）, t 〉 0, and the function R satisfies a nonlinear integrodifferential equation containing u. Under some general conditions, we establish global existence of transient solutions, unique existence of a stationary solution, and convergence of transient solutions toward the stationary solution as t →∞.     

STOCHASTIC HEAT EQUATION WITH FRACTIONAL LAPLACIAN AND FRACTIONAL NOISE:EXISTENCE OF THE SOLUTION AND ANALYSIS OF ITS DENSITY

In this paper we study a fractional stochastic heat equation on R~d(d≥1) with additive noise ?/?t u(t,x) = Dα/δu(t,x) + b(u(t,x)) +W~H(t,x) where D α/δ is a nonlocal fractional differential operator and W~H is a Gaussian-colored noise. We show the existence and the uniqueness of the mild solution for this equation. In addition,in the case of space dimension d=1,we prove the existence of the density for this solution and we establish lower and upper Gaussian bounds for the density by Malliavin calculus.     

Nonradial Entire Large Solutions of Semilinear Elliptic Equations

We consider the problem of whether the equation △u = p（x）f（u） on RN, N ≥ 3, has a positive solution for which lim ｜x｜→∞（x） = ∞ where f is locally Lipschitz continuous, positive, and nondecreasing on （0,oo） and satisfies ∫1∞[F （t）]^- 1/2dt = ∞ where F（t） = ∫0^tf（s）ds. The nonnegative function p is assumed to be asymptotically radial in a certain sense. We show that a sufficient condition to ensure such a solution u exists is that p satisfies ∫0∞ r min｜x｜=r P （x） dr = ∞. Conversely, we show that a necessary condition for the solution to exist is that p satisfies ∫0∞r1＋ε min ｜x｜=rp（x）dr =∞ for all ε〉0.     

Existence, Uniqueness and Blow-Up Rate of Large Solutions of Quasi-Linear Elliptic Equations with Higher Order and Large Perturbation

We establish the existence, uniqueness and the blow-up rate of the large positive solution of the quasi-linear elliptic problem -△pu=λ（x）u^θ-1-b（x）h（u）, in Ω,with boundary condition u = ＋∞ on δΩ, where Ω R^N （N≥2） is a smooth bounded domain, 1 〈 p 〈∞ λ（·） and b（·） are positive weight functions and h（u） ～ uq-1 as u → ∞. Our results extend the previous work [Z. Xie, J. Diff. Equ., 247 （2009）, 344-363] from case p = 2, λ is a constant and θ = 2 to case 1 〈 p 〈∞, A is a function and 1 （ 0 〈 θ 〈q 〉 p）; and also extends the previous work [Z. Xie, C. Zhao, J. Diff. Equ., 252 （2012）, 1776-1788], from case A is a constant and θ = p to case λ is a function and 1 〈 θ 〈 q （ 〉 p）. Moreover, we remove the assumption of radial symmetry of the problem and we do not require h（·） is increasing.     

具$p$-Laplace 算子的三点边值问题正解的存在性

本文用锥上的Krasnoselskii’s不动点定理研究了具有p-Laplace算子的三点边值问题:其中0<α,β<1,0<η<1且(?)p(z)=|x|p-2z,p>1．在f满足一定的增长条件下,得到方程正解的存在性．作为应用,给出两个例子．     

非线性边界条件下二阶奇异差分方程的正解

本文我们考虑如下二阶奇异差分边值问题\begin{equation*}\begin{cases}-\Delta^{2} u(t-1)=\lambda g(t)f(u) ,\ t\in [1,T]_\mathbb{Z},\\u(0)=0,\\ \Delta u(T)+c(u(T+1))u(T+1)=0,\end{cases}\end{equation*}正解的存在性. 其中, $\lambda>0$, $f:(0,\infty)\rightarrow \mathbb{R}$ 是连续的,且允许在~$0$ 处奇异.通过引入一个新的全连续算子, 我们建立正解的存在性.     

一类二阶常微分方程边值问题的无穷多个解

利用变分原理和Z2不变群指标研究了二阶常微分方程边值问题{u″(t)-u(t) f(t,u(t))=0,0＜t＜1,u′(0)=0,α1u(1) u′(1)=0,(其中α1＞-1/2).得出了这类方程存在无穷个解的充分条件.     

一类多点边值问题的可解性

This paper deals with the existence of solutions for the problem
{（Фp（u′））′=f（t,u,u′）,t∈（0,1）,
u′（0）=0,u（1）=∑i=1^n-2aiu（ηi）,
where Фp（s）=｜s｜^p-2s,p〉1.0〈η1〈η2〈…〈ηn-2〈1,ai（i=1,2,…,n-2）are non-negative constants and ∑i=1^n-2ai=1.Some known results are improved under some sign and growth conditions. The proof is based on the Brouwer degree theory.     

An integral boundary value problem of conformable integro-differential equations with a parameter

In this article, we consider some properties of positive solutions for a new conformable integro-differential equation with integral boundary conditions and a parameter $$ \left\{ \begin{array}{l} T_{\alpha}u(t)+\lambda f(t,u(t),I_{\alpha}u(t))=0,t\in[0,1],\u(0)=0,u(1)=\beta\int_{0}^{1}u(t)dt ,\beta\in[\frac 32,2), \ \end{array}\right.\nonumber $$ where $\alpha\in(1,2]$, $\lambda$ is a positive parameter, $T_{\alpha}$ is the usual conformable derivative and $I_{\alpha}$ is the conformable integral, $f:[0,1]\times\mathbf{R^{+}}\times\mathbf{R^{+}}\rightarrow \mathbf{R^{+}} $ is a continuous function, where $\mathbf{R^{+}}=[0,+\infty)$. We use a recent fixed point theorem for monotone operators in ordered Banach spaces, and then establish the existence and uniqueness of positive solutions for the boundary value problem. Further, we give an iterative sequence to approximate the unique positive solution and some good properties of positive solution about the parameter $\lambda$. A concrete example is given to better demonstrate our main result.     

奇异非线性Sturm-Liouville边值问题正解的全局结构

该文利用拓扑方法讨论一类非线性Sturm-Liouville边值问题\[\left\{\begin{array}{lcl}-u'=\lambda f(x, u),\\\alpha_0 u(0)+\beta_0 u'(0)=0,\ \ \alpha_1 u(1)+\beta_1 u'(1)=0;\end{array}\right.\]作者在非线性项不奇异和奇异两种情况下研究了上述问题正解解集的全局结构,在非线性项$f$不满足条件f(x,u)≥0(u≥0)时 获得了正解的存在性.     

一类非线性$m$-点边值问题的正解

应用锥理论和不动点指数方法,在与相应线性算子的第一特征值相关的条件下,得到了下述非线性二阶常微分方程m-点边值问题{u"(t) a(t)u' b(t)u h(t)f(u(t))=0,0＜t＜1,u'(0)=0,u(1)=m-2∑i=1αiu(ξi).的正解,改进了相关文献中的结论.     

一类三点边值问题正解的存在性

在与线性问题第一特征值相关的条件下,通过应用不动点指数理论讨论了三点边值问题u″ 9(t)f(u)=0,t∈(0,1),u′(0)=0,u(1)=αu(η)正解的存在性,这里η∈(0,1),α∈R且0<α<1.本文结果推广和改进了文献[1]的主要结论.     

Uniqueness of solutions for an integral boundary value problem with fractional $q$-differences

This paper deals with uniqueness of solutions for integral boundary value problem$\left\{\begin{array}{l}(D_q^{\alpha}u)(t)+f(t, u(t))=0,\ \ \ t\in(0,1),\ u(0)=D_qu(0)=0,\ \ u(1)=\lambda\int_0^1u(s){\mbox d}_qs, \end{array}\right.$ where $\alpha\in(2,3]$, $\lambda\in (0,[\alpha]_q)$, $D_q^{\alpha}$ denotes the $q$-fractional differential operator of order $\alpha$. By using the iterative method and one new fixed point theorem, we obtain that there exist a unique nontrivial solution and a unique positive solution.